The Expression $\frac{\log \frac{1}{3}}{\log 2}$ Is The Result Of Applying The Change Of Base Formula To A Logarithmic Expression. Which Could Be The Original Expression?A. $\log _{\frac{1}{5}} 2$ B. $\log _{\frac{1}{2}} 3$

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Introduction

The change of base formula is a fundamental concept in mathematics, particularly in the realm of logarithms. It allows us to express a logarithmic expression in terms of a different base, making it easier to work with and manipulate. In this article, we will delve into the expression log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2} and explore which original logarithmic expression could have resulted in this transformation.

Understanding the Change of Base Formula

The change of base formula states that for any positive real numbers aa, bb, and cc, where c≠1c \neq 1, the following equation holds:

log⁑ab=log⁑cblog⁑ca\log_a b = \frac{\log_c b}{\log_c a}

This formula enables us to change the base of a logarithmic expression from aa to cc, making it easier to work with and manipulate.

Applying the Change of Base Formula

Let's apply the change of base formula to the expression log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2}. We can rewrite the expression as:

log⁑13log⁑2=log⁑c13log⁑c2\frac{\log \frac{1}{3}}{\log 2} = \frac{\log_c \frac{1}{3}}{\log_c 2}

where cc is the new base.

Original Expression: log⁑152\log _{\frac{1}{5}} 2

One possible original expression that could have resulted in the transformation log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2} is log⁑152\log _{\frac{1}{5}} 2. Let's apply the change of base formula to this expression:

log⁑152=log⁑2log⁑15\log _{\frac{1}{5}} 2 = \frac{\log 2}{\log \frac{1}{5}}

Using the property of logarithms that log⁑1x=βˆ’log⁑x\log \frac{1}{x} = -\log x, we can rewrite the expression as:

log⁑2log⁑15=log⁑2βˆ’log⁑5\frac{\log 2}{\log \frac{1}{5}} = \frac{\log 2}{-\log 5}

Now, let's apply the change of base formula to the expression log⁑2βˆ’log⁑5\frac{\log 2}{-\log 5}:

log⁑2βˆ’log⁑5=log⁑c2βˆ’log⁑c5\frac{\log 2}{-\log 5} = \frac{\log_c 2}{-\log_c 5}

where cc is the new base.

Original Expression: log⁑123\log _{\frac{1}{2}} 3

Another possible original expression that could have resulted in the transformation log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2} is log⁑123\log _{\frac{1}{2}} 3. Let's apply the change of base formula to this expression:

log⁑123=log⁑3log⁑12\log _{\frac{1}{2}} 3 = \frac{\log 3}{\log \frac{1}{2}}

Using the property of logarithms that log⁑1x=βˆ’log⁑x\log \frac{1}{x} = -\log x, we can rewrite the expression as:

log⁑3log⁑12=log⁑3βˆ’log⁑2\frac{\log 3}{\log \frac{1}{2}} = \frac{\log 3}{-\log 2}

Now, let's apply the change of base formula to the expression log⁑3βˆ’log⁑2\frac{\log 3}{-\log 2}:

log⁑3βˆ’log⁑2=log⁑c3βˆ’log⁑c2\frac{\log 3}{-\log 2} = \frac{\log_c 3}{-\log_c 2}

where cc is the new base.

Conclusion

In conclusion, the expression log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2} is the result of applying the change of base formula to a logarithmic expression. The two possible original expressions that could have resulted in this transformation are log⁑152\log _{\frac{1}{5}} 2 and log⁑123\log _{\frac{1}{2}} 3. Both expressions involve changing the base of a logarithmic expression, making it easier to work with and manipulate.

Final Thoughts

The change of base formula is a powerful tool in mathematics, particularly in the realm of logarithms. It allows us to express a logarithmic expression in terms of a different base, making it easier to work with and manipulate. By understanding the change of base formula and how to apply it, we can solve complex logarithmic expressions and uncover the underlying structure of mathematical problems.

References

Additional Resources

Introduction

In our previous article, we explored the expression log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2} and discovered that it is the result of applying the change of base formula to a logarithmic expression. We also examined two possible original expressions that could have resulted in this transformation: log⁑152\log _{\frac{1}{5}} 2 and log⁑123\log _{\frac{1}{2}} 3. In this article, we will answer some frequently asked questions about the expression log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2} and provide additional insights into the change of base formula.

Q&A

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to express a logarithmic expression in terms of a different base. It is given by the equation:

log⁑ab=log⁑cblog⁑ca\log_a b = \frac{\log_c b}{\log_c a}

where aa, bb, and cc are positive real numbers, and c≠1c \neq 1.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, you need to identify the base of the logarithmic expression you want to change. Then, you need to choose a new base cc and rewrite the expression using the formula:

log⁑ab=log⁑cblog⁑ca\log_a b = \frac{\log_c b}{\log_c a}

Q: What are some common applications of the change of base formula?

A: The change of base formula has many applications in mathematics, particularly in the realm of logarithms. Some common applications include:

  • Changing the base of a logarithmic expression to make it easier to work with and manipulate
  • Simplifying complex logarithmic expressions
  • Solving logarithmic equations and inequalities
  • Understanding the properties of logarithms and their relationships with other mathematical functions

Q: Can I use the change of base formula to change the base of any logarithmic expression?

A: Yes, you can use the change of base formula to change the base of any logarithmic expression, as long as the base is not equal to 1. However, you need to be careful when choosing the new base cc, as it can affect the properties of the logarithmic expression.

Q: How do I choose the new base cc?

A: When choosing the new base cc, you need to consider the properties of the logarithmic expression you want to change. You can choose a base that is easy to work with, such as the natural logarithm (base ee) or the common logarithm (base 10). Alternatively, you can choose a base that is related to the original base, such as a base that is a power of the original base.

Q: Can I use the change of base formula to change the base of a logarithmic expression with a negative exponent?

A: Yes, you can use the change of base formula to change the base of a logarithmic expression with a negative exponent. However, you need to be careful when applying the formula, as the negative exponent can affect the properties of the logarithmic expression.

Q: How do I apply the change of base formula to a logarithmic expression with a negative exponent?

A: To apply the change of base formula to a logarithmic expression with a negative exponent, you need to rewrite the expression using the property of logarithms that log⁑1x=βˆ’log⁑x\log \frac{1}{x} = -\log x. Then, you can apply the change of base formula as usual.

Conclusion

In conclusion, the expression log⁑13log⁑2\frac{\log \frac{1}{3}}{\log 2} is the result of applying the change of base formula to a logarithmic expression. The change of base formula is a powerful tool in mathematics, particularly in the realm of logarithms. By understanding the change of base formula and how to apply it, you can solve complex logarithmic expressions and uncover the underlying structure of mathematical problems.

Final Thoughts

The change of base formula is a fundamental concept in mathematics, and it has many applications in the realm of logarithms. By mastering the change of base formula, you can solve complex logarithmic expressions and gain a deeper understanding of the properties of logarithms.

References

Additional Resources